Text Set #1 - Probability

General Resources pertaining to the many aspects and figures of Probability:

Books

• Leonhard Euler and the Bernoullis:  Mathematicians from Basel
• MBW Tent
• ages 12 and up

This book is a collection of tales about men who were integral in the creation of the mathematical principles in Probability.  This book is a fictional dialogue that is actually historically accurate.  Its contents give students a firm basis of life in the 16th and 17th centuries and tell of how these men discovered their name-bearing theorems.

• A Very Improbable Story:  A Math Adventure
  
• Edward Einhorn & Adam Gustavson
• ages 7 and up

This book is a picture book that details a young boy's day and the many challenging and improbable things he must complete in order to get a cat off his head.  The boy must complete games of daunting probability from the time he brushes his teeth until late in the evening.  The book contains hilarious depictions of the scenes and details the mathematics used in each situation.

• The Cartoon Guide to Statistics
• Larry Gonick & Woollcott Smith
• ages 14 and up

This book has an uncanny ability to provide informational and often humorous depictions of instances in which probability and statistics are prevalent in the real world.  The material covered in this work are of higher level, yet are written in a manner understandable by early high school students.  Theories of probability are easily portrayed in this work.

• Probability Demystified
• Allan Bluman
• Ages 14 and up

This book is another example of a work that presents the difficult material of a probability course in a manner easily understood by beginner students.  The work contains the major theorems of the course as well as topics that are often taken for granted (sample space, events, etc).  The book contains all of these and up to the many higher-level aspects of probability.  

• Math Made Easy:  Combinations and Permutations
• Dawn LaBuy-Brockett
• Ages 5 and up

This book teaches the many aspects of combinations and permutations, from the basics to the more in-depth concepts that entail.  The work contains easily understood explanations and pictures detailing factorials, groups, ordered sets, and more.  The pictures are clear and concise and provide a great "hands up" for the reader.

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Websites

• Tales of Statisticians:  Jacob Bernoulli
• http://www.umass.edu/wsp/statistics/tales/bernoulli.html
• Ages 18 and up

An in-depth look at the life and accomplishments of one of math's greatest minds.  Bernoulli is the mind behind the Bernoulli Trials, an integral aspect of the probability curriculum.  The work also notes the other, maybe less-known discoveries of Jacob Bernoulli.

• Better Explained:  Easy Permutations and Combinations
• http://betterexplained.com/articles/easy-permutations-and-combinations/
• Ages 13 and up

This website is perfect for the beginner probability student as it details, in layman's terms, how to calculate permutations and combinations.  The terminology used in this site is very simple yet very forth.  It tells the reader exactly what is needed in a very direct manner.  The site also has multiple images that depict the situations perfectly.

• Probability Test 1 Flashcards
• http://www.cram.com/cards/probability-test-1-379658
• Ages 13 and up

Cram.com is a site in which the general public can post files for subjects across the board.  This particular file is a set of flashcards that detail multiple key vocabulary terms for an introductory probability course.    The site allows students to cycle through the "front" of the flashcards to quiz themselves and allows the students to flip from front to back when studying.

• Experimental Probability - Horse Basketball Game
• http://math.about.com/od/probability/a/Prob_JL.htm
• Ages 11 and up

This site is actually an in-depth example of experimental probability.  The girls in this situation are vying to win at "Horse".  The students are then given the outcomes of ten games and are asked to derive the probability of many other situations.  This would be perfect to provide a real world application of the concept.

• Probability Through Problems:  A New Approach to Teaching Probability
• http://nrich.maths.org/9837
• for Teachers

In this website, readers are given multiple resources through which we can teach probability to our students.  There are many lessons that teachers can freely access and utilize.  There is also a great description of Nrich's ideal classroom in which a problem is the basis for a classroom and not a technique.

• MathWords:  Bernoulli Trials
• http://www.mathwords.com/b/bernoulli_trials.htm
• Ages 13 and up

Mathwords is a site that has an incredibly extensive selection of mathematics vocabulary.  Each term has its own page that delves deep into its meaning.  This particular page pertains to Bernoulli Trials.  The site gives many examples in which a Bernoulli Trial is prevalent and also gives a great definition of the term.  The page also defines what each variable in the equation means, in easy to understand terminology.

• Stat Trek:  Binomial Distribution
• http://stattrek.com/probability-distributions/binomial.aspx
• Ages 17 and up

This site is a deeper look into Bernoulli Trials and Binomial Distributions.  Stat Trek is built for upper high school to collegiate level students and the terminology used is equivalent in level.  The students are subject to a more academic-specific vocabulary set in the description, but it is still an easy to understand site with tons of information.

• Math Forum:  As Dr Math FAQ - Probability in the Real World
• http://mathforum.org/dr.math/faq/faq.prob.world.html
• Ages 16 and up

This particular portion of the Math Forum details "Probability in Our Lives".  It is a good scope into how the basic ideals of probability and statistics are evident in many everyday activities.  One of the best instances outlined on the website is in weather forecasting.  Though weather probability is much more than what is detailed on the math forum, we are still given a great look into how the job is done.  There are also many other instances that bring about many other aspects of probability.

• EMathZone:  Sample Space
• http://www.emathzone.com/tutorials/basic-statistics/sample-space.html
• Ages 12 and up

This site on the EMathZone is dedicated to one of the most basic ideas in all of probability:  the sample space.  Without a sample space, however, there are no odds or likelihoods.  This site gives great and very simple definition of a sample space.  It also gives good, concrete examples of sample spaces in multiple formats (Bernoulli, binomial, etc).  The spaces are even depicted as images so the reader is given multiple medias from which to learn.

• WikiHow:  How to Calculate an Expected Value
• http://www.wikihow.com/Calculate-an-Expected-Value
• Ages 15 and up

WikiHow somewhat receives a bad rap because it can be edited by any person imaginable (credentials or any Joe off the street).  However, this particular wiki is great because it is accurate.  The wiki is also very easy to follow along and easy to learn from.  The only drawback is the author's decision to use higher-level vocabulary to describe the action necessary.  The wiki also has great pictures that enable the reader to see the discovery of the expected value.  The situation is also a game, so the reader is more apt to follow to see if he/she should even play the game (applicable to their lives).

Bromley Article

The Bromley article was a bit different than previous readings.  This article detailed certain facts about vocabulary and the English language in general.  I found it incredibly interesting to read about the language.  I knew English was a large and expansive language, but I did not know it was THAT much larger than French or German.  I knew that a bunch of words had multiple meanings, but 70% of our most used words?  I was astonished by that figure.  However, the one fact that took me aback was the one that stated that the rules of English are more consistent and simpler than other languages.  I always thought English was the toughest language to learn because of the nuances and rules.  Apparently not.  I guess I could see how this is the case, especially when compared to Chinese.  All in all, I thought it was cool to learn more about my language.

In regards to my classroom, the bullet point that I noticed was number four, "Words are learned because of associations that connect the new with the known."  I think, especially in math, students must be able to draw connections between words to understand the vocabulary.  There are multiple roots and affixes of math vocabulary that are prevalent in many, many words.  As I have said before in this blog, I can utilize that aspect of math vocabulary via word clouds or other strategies.  Focusing on those common words/parts of words can certainly build student vocabulary.

Building Vocabulary

Finally a chapter in which I looked at it and said, "I KNOW I'll find something useful in here!"  Teaching math deals with a lot of building vocabulary.  There are thousands upon thousands of terms that students must know in order to really be successful.  Words that they have learned from day one will impact their performance in high school mathematics classes.  Geometry requires the learning of  new and integral terms seemingly every lesson.  Vocabulary is HUGE in math.  Now, how can I help the students learn these terms?

Just by looking at the strategies from BBR, I was instantly drawn to the R^3 Strategy.  The math in me saw exponents and lit up.  But it was not to be.  I think R^3 would be very effective in history and science classes, but there isn't quite enough reading opportunities in math to really utilize R^3.

The strategy I like, though, is Morphologia.  I like this strategy because math uses old terms and Latin and Greek roots.  Though Morphologia does not require the study of the roots by the students.  Rather, they are exposed to morphemes or parts of the word that are easier to understand.  In geometry, each of the shapes have morphemes that help students understand the entire word's meaning.  Words like pentagon and hexagon are perfect examples.  Penta-, hexa-, and -gon are morphemes that draw to five, six, and sides, respectively.    Using words like the aforementioned helps students understand the strategy.  Using morphologia to understand complex math terms can help the students break it down and understand a general meaning.

WA

D&Z Ch 11

In this chapter, D & Z outline many ways in which teachers can aid struggling readers.  One of which is a constant strategy in every topic:  Build Supportive Relationships.

No matter what the desired outcome is, building solid relationships with our students is a way to increase the likelihood of success.  Teachers should put forth the effort so that students understand how important their success in reading really is to us.  We should make them understand that we really care about them.

One thing I must add, though.  Each of these strategies should be used for the entire class.  Every student would benefit from teachers modeling thoughtful reading.  Every student would improve with the aid of other materials such as tapes and supporting literature.  We must determine what frequency we use them with the higher reading levels.  But each strategy would benefit every student.

BBR Ch 6 - Quick Writes

In this chapter, BBR outlines a few strategies to help improve student reading comprehension.  I have learned about each of these at some point or another in my collegiate career and some moreso than others.  I have used KWL's in my class before and I love how the strategy utilizes its three components to develop a solid pre- and post-reading assignment.  

I can see how these can be used in mathematics, also.  Graphic Organizers can be utilized in geometry for area and volume facts about different figures.  Think alouds are used when completing tasks in common core.  The students have to discuss and find the best ways to complete a problem and discuss the multiple ways that can be accomplished.

The one strategy that piqued my interest when reading, though, were Quick Writes.  BBR states that they can be used "as mirror assignments for state reading assessments," but I can see them used as the same for the mathematics portion of the exams.  With Common Core on the horizon (PARCC), students will be asked to annotate their work, telling how they came up with answers.  I can see quick writes being used as such.  Math teachers drill how one procedure or concept is different from another; quick writes can be assigned to check the understanding of the students while preparing them for the PARCC.  An example question for a quick write could be, "Describe arithmetic and geometric series.  How will their graphs differ?"  The students would have to draw information from the lesson and generate a few sentences detailing arithmetic and geometric series.  IN such a manner, I think quick writes would be perfect in math class.

WA